Average Error: 0.1 → 0.1
Time: 39.5s
Precision: 64
Internal Precision: 128
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{(-9 \cdot \left(\frac{1.0}{3.0}\right) + \left(a \cdot 9\right))_*}} \cdot \left(a - \frac{1.0}{3.0}\right)\]

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
  2. Using strategy rm

  3. Applied *-un-lft-identity0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) \cdot \color{blue}{\left(1 \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\right)}\]

  4. Applied associate-*r*0.1

    \[\leadsto \color{blue}{\left(\left(a - \frac{1.0}{3.0}\right) \cdot 1\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)}\]

  5. Simplified0.1

    \[\leadsto \left(\left(a - \frac{1.0}{3.0}\right) \cdot 1\right) \cdot \color{blue}{\left(1 + \frac{rand}{\sqrt{(-9 \cdot \left(\frac{1.0}{3.0}\right) + \left(a \cdot 9\right))_*}}\right)}\]
  6. Using strategy rm

  7. Applied distribute-lft-in0.1

    \[\leadsto \color{blue}{\left(\left(a - \frac{1.0}{3.0}\right) \cdot 1\right) \cdot 1 + \left(\left(a - \frac{1.0}{3.0}\right) \cdot 1\right) \cdot \frac{rand}{\sqrt{(-9 \cdot \left(\frac{1.0}{3.0}\right) + \left(a \cdot 9\right))_*}}}\]

  8. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{(-9 \cdot \left(\frac{1.0}{3.0}\right) + \left(a \cdot 9\right))_*}} \cdot \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))